Investgating resistivity - Planning and Implementing

Three pairs of values for voltage and current for each length are measured to make the value for R obtained from each length more reliable. If the wire obeys Ohm’s Law, no matter the voltage and current, the value obtained for resistance from the formula R=V/I, should be the same, meaning three values for one resistance can be obtained, which, if they show to be equal to within experimental error, improves the reliability of the value of R obtained. However, Ohm’s Law also states that for V=IR to be true, the external conditions must be the same. As a wires current increases, so does its temperature. If the temperature increases greatly, this will cause the resistance to increase, because as a metal’s temperature increases, the electrons and positive ions in the metal lattice gain more energy, causing the positive ions to vibrate more and the electrons to move in their random direction faster, causing more collisions between the drifting electrons and positive ions, hence increasing the resistance. So, to allow the reliability of R to be ensured by calculating it several times, the temperature change of the wire must be kept to a minimum, so that Ohm’s Law is obeyed. This means the current must be kept below 1A.

Using the experimental data

Using V=IR, calculate the resistance for each pair of voltages and currents. Compare the resistances obtained from each length. At any one length, the resistances should be equal to within experimental error. Calculate the mean resistance for each length, ignoring any values that appear anomalous. Then plot a graph of the mean resistance in Ohms against the length in metres. Find the gradient, using change in resistance divided by change in length over at least 8cm of graph, which represents ρ/A. Then use the diameter of the wire obtained from the micrometer to calculate the cross-sectional surface area of the wire from the formula A=π(½D)2. Use ρ/A=gradient, substituting in A and the gradient to find a value for ρ. Next, best fit lines of maximum possible and minimum possible gradient should be drawn, and the gradients of these line found and used to calculate alternative values for ρ. The difference between these maximum and minimum values for ρ should be found, so that one can state that from the graph ρ = a+/-b Ωm, where a is the value for the resistivity from the line of best fit, and b is the largest difference in the value of resistivity between the minimum/maximum value and a. A graph should also be drawn using excel, and the value obtained here compared with that obtained from the hand-drawn lines of best fit.

Safety

The current should be kept below 1A to avoid the wire getting hot, which could cause burns. Safety goggles should be worn, as in the event that the wire snaps one’s eyes could be damaged. Given the electricity involved, the experiment should be kept away from water, or other liquid conductors.

Hypothesis

Each value of R obtained from a certain length will be equal within the limits of experimental error. When R is plotted against l, the line of best fit will be a straight line through the origin, demonstrating direct proportion. The gradient should represent ρ/A.

Variables

The independent variable in the experiment will be the length of constanton wire in the circuit. This variable will be changed, and the dependent variables of voltage and current will be recorded. The resistance in the rest of the circuit will be altered using the variable resistor, allowing multiple values of the current and voltage through any length of wire to be taken. A variable to be kept constant is the temperature of the wire, to allow these different values of current and voltage to be recorded without changing R. This can be achieved by keeping the current below 1A, which will help prevent the wire from heating up. Also the cross sectional area of the wire will be kept constant, by using the same piece of wire to test each length, and measuring its diameter using the micrometer in several places, to reduce the chance of there being unnoticed discrepancies in its width.  Using the same wire for each length will also ensure that ρ remains constant as well.

Implementing

Precision of readings

The voltmeter and ammeter measure to the nearest 0.005, hence all values for voltage and current are recorded to two decimal places. Resistances are also given to two decimal places, as given the accuracy of the voltage and current the resistance is calculated with, further decimal places could not be supported. Length is measured to the nearest mm, so these values in metres are given to three decimal places. Diameter is measured to the nearest 0.005 mm, so it is given in millimetres to two decimal places.

Significant sources of error

The length of wire, current and voltage, and the wire’s diameter are measured. The length of the wire has the smallest error, as the smallest value is 0.1m, and it is measured to the nearest millimetre. Percentage error, at its highest, was therefore

0.5% was not a particularly significant error, especially given this error only occurs once, at 10cm, and the other errors at greater lengths are smaller.

For voltage and current, the maximum error is of 0.005 when the voltage or current is at its lowest. For voltage the highest percentage error was therefore

And for current the highest percentage error was therefore

 

These errors are quite significant. If they were any greater the value for the resistance calculated from these currents and voltages, and therefore the value of resistivity calculated from that resistance, might be too unreliable. To prevent this, the current and voltages must not be allowed to fall below 0.2, and be kept as high wherever possible.

The error in the diameter is, at the very most, 0.005mm. This means the error in the diameter is

However, this value is squared in calculating the area, which doubles this error, increasing it further to 4%, a very significant amount. To keep this error to a minimum, the thickest wire possible was used, while still being thin enough to be easily straightened and kept taut. If it were much thinner, the magnitude of possible error would make it more difficult to produce a reliable and realistic value for ρ.

Calculations

The formula

was used to calculate the resistance for each pair of voltages and currents. Then, the mean of the three values of R obtained for each length was calculated, by finding the sum of the three values then dividing it by three. The mean of the three measurements from the micrometer was also taken, to find the mean diameter.

Results obtained: